Bounds for Toader Mean in Terms of Arithmetic and Second Seiffert Means

Zai-Yin He; Yue-Ping Jiang; Yu-Ming Chu

doi:10.26713/cma.v10i3.1200

Authors

Zai-Yin He College of Mathematics and Econometrics, Hunan University, Changsha 410082
Yue-Ping Jiang College of Mathematics and Econometrics, Hunan University, Changsha 410082
Yu-Ming Chu Department of Mathematics, Huzhou University, Huzhou 313000

DOI:

https://doi.org/10.26713/cma.v10i3.1200

Keywords:

Toader mean, Second Seiffert mean, Arithmetic mean

Abstract

In the article, we prove that the double inequalities
$\begin{aligned} α_{1} T (a, b) + (1 - α_{1}) A (a, b) < T D (a, b) < β_{1} T (a, b) + (1 - β_{1}) A (a, b), \\ T^{α_{2}} (a, b) A^{1 - α_{2}} (a, b) < T D (a, b) < T^{β_{2}} (a, b) A^{1 - β_{2}} (a, b) \end{aligned}$
hold for all $a, b > 0$ with $a \neq b$ if and only if $α_{1} \leq 3 / 4$ , $β_{1} \geq 1$ , $α_{2} \leq 3 / 4$ and $β_{2} \geq 1$ ,
where $A (a, b)$ , $T D (a, b)$ and $T (a, b)$ are the arithmetic, Toader and second Seiffert means of $a$ and $b$ , respectively.

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Bounds for Toader Mean in Terms of Arithmetic and Second Seiffert Means

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